Question

The following information is given about options on the stock of a certain company.

S₀ = 23                   X = 20

r_c = 0.09                 T = 0.5

s² = 0.15

No dividends are expected.

1)What is the value of the call option based on the Black and sholes Merton model?

2) If the Stock is distributing dividends of $0.85 with 12 days ex-dividends day what is the value of the call?

3) what is the Delta of the option, what about the Gamma ?

4) if the Volatility increased to 0.45 what is the value of the call?

Answer 1

Ans a)

Value of the call option based on the Black and sholes Merton model:

The Black–Scholes-Merton formula for value for a European call option:

C(S₀​,t)=S_0​N(d₁​)−Xe^−r(t)N(d₂​),

where

· S0​ is the stock price; = 23

· C(S0​,t) is the price of the call option as a formulation of the stock price and time;

· X is the exercise price; = 20

· (T) is the time to maturity. Generally, this is represented in years. = 0.50

· r = interest rate = 0.09

· σ²= 0.15

Putting above values in formula for d₁ gets d₁ = 0.5229 and d₂ = 0.2490

From Z table it can be seen that,

N(d₁​) = N(0.5229) = 0.6995

N(d2) = N(0.2490) = 0.5983

So putting above values in the formula for Call option gives

C(S₀​,t)=S₀​N(d₁​)−Xe^−r(t)N(d₂​)

C(23​,0.5) = 23 * 0.6995 – 20 e^ ^{- 0.09*0.5} * 0.5983

C(23​,0.5) = $4.65

Ans. b)

If the Stock is distributing dividends of $0.85 with 12 days ex-dividends day what is the value of the call:

= Value of the Call without dividend – Present value of the dividend

= $4.65 – ($0.85)^^{1+ (.09*12/365)}

⁼ $4.65 - $0.8475

= $3.8025

Ans. c)

What is the Delta of the option, what about the Gamma ?

Based on the Black and sholes Merton model ,

Delta of the option = 0.793

Gamma of the option = 0.0466

Ans. d)

If the Volatility increased to 0.45 the value of the call

In the formulae as mentioned at solution at Ans. a) above, if the volatility is replaced by 0.45, the value of Call option would be $6.06.